It's one of the odd things about West Berlin that they never built a metro connection to Tegel Airport.
Several of us who went to Berlin pre (and post) reunification seem to have been fascinated with it.
Tegel was not the airport of West Berlin until 1975, when the flights all moved over from Tempelhof. There was one airport in each sector, including in the East (not the current Schonefeld airport, which was actually just outside East Berlin, in GDR proper, giving rise to a further set of bureaucratic conundrums if you wanted to use it). Notably, not all but many of these issues could be overcome if you worked with the system, and the population on both sides seemed able to do so - although not the impression given by politicians, from either camp.
The "duty free" Intershop kiosks on Friedrichstr platforms sold to "westerners" not entering East Berlin, as it was also an interchange between west side S-Bahn (elevated) and U-Bahn (tunnel) without entering East Berlin. The key thing about them was they only accepted western currency (principally Deutschmarks), it was a means of the state collecting this. There was also surprisingly more trade between east and west than might be expected. The water supply in the west wholly came from the GDR.
The Intershop staff, like others who interchanged with the West, were appropriately selected (like the many GDR seamen) from those with families and children who remained in the GDR, and thus were unlikely to want to escape. Most were within a few years of retirement, after which you could more readily cross to the west (escaping retirees would save the GDR pension costs). The principal item sold was cigarettes, at slightly below western prices; it was called "duty free" but of course the GDR government collected it all.
I never quite got the position of the staff on the isolated S-Bahn network in West Berlin, operated by the DR (GDR). I believe they were westerners, but paid by the east, only at the official GDR staff rate and topped up by the West government. I gather wages fell behind and ultimately there was a strike which led in the 1980s to quite an amount of the west side S-Bahn network closing. The movable barriers across the link track, shown in the video linked above, were used to move the trains, owned by the DR, back and forth to the maintenance depot. I read they were propelled to the halfway point by a diesel shunter from the west, and then picked up at the other end by a shunter from the east, so no driver had to cross the border.
You had to exchange 25 (I think) Deutschmarks for GDR Marks at a 1:1 exchange rate each day as you crossed over, actually I used to change more as the better restaurants in East Berlin were as good as anywhere and you could have a very pleasant lunch and dinner there. Transport was ludicrously cheap, I can't remember (someone else will) whether the tram was still 5 Pfennigs, about a penny. No conductor, you just put the money in a box by the door and pressed a handle, whereupon it was displayed with the last few fares through the glass, before falling into the cash vault below with a ker-ching. No tickets. Season ticket holders (monthly only I think) would hold them up high for all the other passengers to see.
East Berlin suburbs, and tramway/S-Bahn etc expanded across the city line into the GDR, completely unmarked, but a day visitor to East Berlin was prohibited from crossing the line.
I have upstairs a book called "Berlin Transit"
https://www.amazon.com/Berlin-Transit-1986-87-J-H-Price/dp/0906273269 , written by J H Price, the longstanding editor of Cooks Continental Timetable, also great tramway buff and editor of Modern Tramway magazine; it was very useful and describes all the various combinations of transport and border crossing, which varied depending on whether you were West German, Allies (so being British an advantage), or third party, including for example the challenges of taking a taxi across the border at a time when many of the West Berlin drivers were Turkish.